In this talk we introduce the class of interval and proper interval graphs and we present some equivalent representations of them. Furthermore, we briefly summarize the optimization problems that can be efficiently solved in these classes of graphs. Finally, we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem.

Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.

This is joint work with Walter Unger.